# Algebra

Document Sample

```					Level E
Algebra
Tidying up terms
www.mathsrevision.com

Number Machine
Working with Expressions
Solving Simple Equations
Solving Harder Equations
Understanding Inequalities
Solving Simple Inequalities

25-Nov-11        Created by Mr.Lafferty Maths Dept
Level E
Starter Questions

2
Q1.         Calculate     66 % of £270
3
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Q2.         Convert 60% to fraction and simplify.

4
Q3.         Convert       to a percentage.
5
Q4.         What is the time difference 09:28 and 10:50

Q5.         The answer to the question is 90. What is the
question.

25-Nov-11                 Created by Mr.Lafferty Maths Dept
Level E
Aim of the Lesson

Tidying Terms
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a+2b+3a+b

1. Tidying up expressions.
2. Show steps when tidying up.

25-Nov-11         Created by Mr.Lafferty Maths Dept
Level E
Tidying Terms

First Row :        +                      +                 =
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2c   +           c          +            4c = 7c

2nd Row :           +                       +               =
25-Nov-11
2d      +          2d           +
Created by Mr.Lafferty Maths Dept
3d   = 7d
Level E
Tidying Terms

3rd Row :          +                       +              =
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3f    +           f          +            2f = 6f
CANNOT
In Total we have                                          TIDY UP
ANYMORE
7c + 7d + 6f
WE CAN ONLY TIDY UP “LIKE TERMS”
25-Nov-11         Created by Mr.Lafferty Maths Dept
Level E
Tidying Terms

WE CAN ONLY TIDY UP “LIKE TERMS”
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Tidy up the following:

Q1. 2x+4x+5y-3y+18 6x+5y-3y+18 = 6x+2y+18

Q2. 4a+3b+5a+6-b                                 9a+3b-b+6 = 9a+2b+6

25-Nov-11                  Created by Mr.Lafferty Maths Dept
Level E
Tidying Terms
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Your turn !

Page 46 Exercise 1
Q1 and Q2
25-Nov-11        Created by Mr.Lafferty Maths Dept
Level E
Starter Questions

Q1.         Tidy up the expression                  2y+3x+4x+10
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2
Q2.         Calculate     of 84
3
Q3.         Round to 1 decimal place.

(a)   2.34 (b)             10.25 (c)            3.23

Q4.         12.5 % as a fraction

25-Nov-11               Created by Mr.Lafferty Maths Dept
Level E
Aim of the Lesson

Number Machine
www.mathsrevision.com

1. To build simple expressions.
2. To UNDO simple expressions.

25-Nov-11         Created by Mr.Lafferty Maths Dept
Level E
Number Machine

To build an expression from the number machine
simply follow the rules given:
www.mathsrevision.com

4a
Multiply
a                                          Add 1    4a+1
By 4

7m
Multiply                       Subtract
7                                                    7m-2
By m                             2

25-Nov-11            Created by Mr.Lafferty Maths Dept
Level E
Number Machine

To build an expression from the number machine
simply follow the rules given:
www.mathsrevision.com

Multiply
20
4
By 5
+3    23
?

Multiple
3b
3                                              +9   3b+9
?
by b

25-Nov-11            Created by Mr.Lafferty Maths Dept
Level E
Number Machine
www.mathsrevision.com

Your turn !

Page 93 Exercise 2
Q1
25-Nov-11        Created by Mr.Lafferty Maths Dept
Level E
Number Machine

To UNDO the number machine
simply do the opposite of the rules given:
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x2          UNDO                 +8       UNDO

Divide by            18
36
?
2
-8          10

25-Nov-11              Created by Mr.Lafferty Maths Dept
Level E
Number Machine

To UNDO the number machine
simply do the opposite of the rules given:
www.mathsrevision.com

x4          UNDO                 -9           UNDO

Divide by            5
20
?
4
+9          14

25-Nov-11              Created by Mr.Lafferty Maths Dept
Level E
Number Machine
www.mathsrevision.com

Your turn !

Page 93 Exercise 2
Q3 and Q4
25-Nov-11        Created by Mr.Lafferty Maths Dept
Level E
Starter Questions

1. Simplify a  4a  13d  11a  a  3d
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2. The ratio of apples to pears is 2 : 5.
If there are 28 pieces of fruit in total.
How many apples are there.

3.         Find HCF of 8 and 18

25-Nov-11              Created by Mr.Lafferty Maths Dept
Level E
Substitution

Learning Intention                                    Success Criteria
www.mathsrevision.com

1. To explain the process of                        1. Understand process of
working out expressions by                          substitution.
substitution method.
2. Solving expressions

showing working.

25-Nov-11                    Created by Mr.Lafferty Maths Dept
Level E
Substitution

1    If a = 3 and b = 9 evaluate
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a+b
Sub in a=3
= 3+ 9                               Sub in b=9

= 12
25-Nov-11         Created by Mr.Lafferty Maths Dept
Level E
Substitution

2           If a = 3 and b = 9 evaluate
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ab
axb                       Remember ab means a x b

Sub in a=3
= 3x9                       Sub in b=9

25-Nov-11   = 27    Created by Mr.Lafferty Maths Dept
Level E
Substitution

3           If a = 3 and b = 9 evaluate
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4b
4xb              Remember 4b means 4 x
b

= 4x9               Sub in b=9

= 36
25-Nov-11           Created by Mr.Lafferty Maths Dept
Level E
Substitution

If a = 5 and b = 6 evaluate
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1. a + b =11                           4. b - a =1

2. ab         =30                      5. 4b      =24

3. 3a        =15                       6. 3ab     =90
25-Nov-11     Created by Mr.Lafferty Maths Dept
Level E
Substitution

4           If a = 5 and b = 3 evaluate
2a + 4b
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= 2xa + 4xb                       Remember 2a means 2 x a

Sub in a=5
= 2x 5 + 4x3                          Sub in b=3

= 10           + 12
= 22
25-Nov-11         Created by Mr.Lafferty Maths Dept
Level E
Substitution

5           If a = 5 and b = 3 evaluate
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a2

= a x a                             Remember a2 means a x a

Sub in a=5
= 5x 5
= 25
25-Nov-11            Created by Mr.Lafferty Maths Dept
Level E
Substitution

6           If a = 15 and b = 3 evaluate
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a  15                          Remember a means a ÷ b
=                                      b
b   3
Sub in
= 5                         a=15
Sub in b = 3
25-Nov-11            Created by Mr.Lafferty Maths Dept
Level E
Substitution

If a = 2 and b = 6 evaluate
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1) 4a + 3b =26 4) 5b – 3a =24
2) a2           =4 5) 4a2                         =16
3) b2          =36 6) b =3
a
25-Nov-11     Created by Mr.Lafferty Maths Dept
Level E
Substitution
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Your turn !

Page 94 Exercise 3

25-Nov-11        Created by Mr.Lafferty Maths Dept
Level E
Starter Questions

1. Simplify 3 x  x  6  x  x  3
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2. The ratio of apples to pears is 3 : 1.
If there are 24 pieces of fruit in total.
How many apples are there.

3.     Find LCM of 6 qand 21

25-Nov-11             Created by Mr.Lafferty Maths Dept
Level E
Equations
Level E Solving Equations

Learning Intention                                     Success Criteria
www.mathsrevision.com

1. To solve simple equations                        1. Know the process of the
using the                                           „Balancing Method‟.
„Balancing Method‟.
2. Solving simple algebraic

equations.

25-Nov-11                    Created by Mr.Lafferty Maths Dept
25-Nov-11                                       Created by Mr Lafferty Maths Dept

Balancing Method
Kirsty goes to the shops every week to buy some potatoes.
She always buys the same total weight.
One week she buys 2 large bags and 1 small bag.
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The following week she buys 1 large bag and 3 small bags.

If a small bags weighs 4 kgs. How
much does a large bag weigh?                                      4
4       4

4

How can we go about
What instrument
solving this using
measures balance
balance ?
25-Nov-11                              Created by Mr Lafferty Maths Dept

Balancing Method
Take a small bag away
from each side.                                        4
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4   4       4
Take a big bag away
from each side.

We can see that a big bag
is equal to
4 + 4 = 8 kg
25-Nov-11                                Created by Mr Lafferty Maths Dept

Balancing Method
What symbol should
we use for the scales
?

Let‟s solve it using maths.
4
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Let P be the weight                           P      P
4
P
4       4
of a big bag.

We know that a small bag = 4

2P + 4 = P + 12                       Subtract 4
-4           -4                 from each side

2P = P + 8                           Subtract P
-P   -P                            from each side

P=8
25-Nov-11                                       Created by Mr Lafferty Maths Dept

Balancing Method
Group of 5 adults and 3 children go to the local swimming.
Another group of 3 adults and 8 children also go swimming.
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The total cost for each group is the same.
A child‟s ticket costs £2.                         2     a

If a child‟s ticket costs £2. How much                                2   a   a
for an adult ticket ?                                             2   2   2 2
a
a   a   2   2   2 2
Let a be the price                                     a   a   2
of an adult ticket.

We know that a child price = £2
25-Nov-11                                 Created by Mr Lafferty Maths Dept

Balancing Method
Subtract 6
Subtract 3a
For balance we    have each side
from
2       a
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2   a   a
5a + 6 = 3a + 16                                   a       2
-6            -6                               a   a   2   2   2 2

a   a   2   2     1
2 2
5a = 3a + 10
-3a   -3a

Divide each
2a = 10                     side by 2
a= 5

Adult ticket price is £5
25-Nov-11                              Created by Mr Lafferty Maths Dept

Balancing Method
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It would be far too time consuming to draw out
the balancing scales each time.

We will now learn how to use the rules for
solving equations.
Level E
Equations
Level E Solving Equations

The method we use to solve equations is
www.mathsrevision.com

The Balancing Method
Write down the opposite of the following :

+             -
opposite is                          - opposite is +
x opposite is ÷                                  ÷ opposite is x
25-Nov-11               Created by Mr.Lafferty Maths Dept
Level E
Equations
Level E Solving Equations

The Balancing Method
www.mathsrevision.com

Subtract
Add 5 to
8 from
each side
Examples                                                 each side

x  5  11                                     x8 8
 x  11  5                                  x  88
 x  16                                      x0
Check to see if correct !                 Check to see if correct !
25-Nov-11            Created by Mr.Lafferty Maths Dept
Level E
Equations
Level E Solving Equations

The Balancing Method
www.mathsrevision.com

Divide
Divide                                  each side
each side                                  by 8
Examples            by 3
3 y  18                                     8m  72
18                                       72
y                                         m
3                                        8
 y6                                      m9
Check to see if correct !                   Check to see if correct !
25-Nov-11            Created by Mr.Lafferty Maths Dept
Level E
Equations
Level E Solving Equations
www.mathsrevision.com

Now try Exercise 4
Ch8 (page 96)

25-Nov-11          Created by Mr.Lafferty Maths Dept
Level E
Starter Questions

1. Simplify 5 y  6 x  7 x  3 x  3 xy
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2. The ratio of apples to pears is 4 : 5.
If there are 72 pieces of fruit in total.
How many apples are there.

3.     Find HCF of 15 and 21

25-Nov-11             Created by Mr.Lafferty Maths Dept
Level E
Equations
Level E Harder Equations

Learning Intention                                    Success Criteria
www.mathsrevision.com

1. To solve harder equations                        1. Know the process of
using the rule                                       „Balancing Method‟.
„Balancing Method‟.
repeatedly.
2. Solving harder algebraic
equations by using rule

repeatedly.

25-Nov-11                    Created by Mr.Lafferty Maths Dept
Level E
Equations
Level E Harder Equations

The rule we use to solve equations is
www.mathsrevision.com

The Balancing Method

Write down the opposite of the following :

+             -
opposite is                         - opposite is +
x opposite is ÷                                 ÷ opposite is x
25-Nov-11              Created by Mr.Lafferty Maths Dept
Level E
Equations
Level E Harder Equations

The Balancing Method
www.mathsrevision.com

Examples                Subtract
1 from
2 x  1  11            each side

 2 x  11  1  10   Divide
each side
 2 x  10             by 2

10
 x   5 Check to see if correct !
25-Nov-11
2          Created by Mr.Lafferty Maths Dept
Level E
Equations
Level E Harder Equations

The Balancing Method
www.mathsrevision.com

Examples                  Add 6 to
each side
4 x  6  30
 4 x  30  6  36        Divide
each side
 4 x  36                  by 4

36
x         9 Check to see if correct !
4
25-Nov-11            Created by Mr.Lafferty Maths Dept
Level E
Equations
Level E Harder Equations
www.mathsrevision.com

Now try Exercise 5
Ch8 (page 97)

25-Nov-11         Created by Mr.Lafferty Maths Dept
Level E
Starter Questions

1. Simplify 7 a  4a  9b  11b  ab  3
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2. The ratio of apples to pears is 6 : 7.
If there are 52 pieces of fruit in total.
How many apples are there.

3.         Find LCM of 4 qand 7

25-Nov-11              Created by Mr.Lafferty Maths Dept
Level E
Inequalities
Level E Solving Inequalities

Learning Intention                                     Success Criteria
www.mathsrevision.com

1. To explain the various                           1. Know rule the 4 symbols for
symbols for inequalities.                           inequalities.

2. Solving simple inequalities

equations.

25-Nov-11                    Created by Mr.Lafferty Maths Dept
Level E
Inequalities
Level E Solving Inequations
The equations you were solving were called
equations because they contained an “=“ sign.
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There are 4 other Mathematical symbols
They are called inequalities :

      greater than                   greater than or equal to

      less than                       less than or equal to

25-Nov-11            Created by Mr.Lafferty Maths Dept
Level E
Inequalities
Level E Solving Inequalities
Arrow ALWAYS
points to the
Examples : Write in words what each mean.
smallest number
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5>3        means 5 is greater than 3
-7 < -6       means -7 is less than -6
x≥2        means x can take the value
2 or greater
y ≤ -5     means y can take the value
-5 or less
25-Nov-11          Created by Mr.Lafferty Maths Dept
Level E
Inequalities
Level E Solving Inequalities

Examples : Write out an inequality for the numbers
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(a)   14 and 6                        14 > 6

(b)   -2 and 1                        -2 < 1

(c)   -3 and -7                      -3 > -7

(d)   34 and -35                   34 > -35

25-Nov-11                Created by Mr.Lafferty Maths Dept
Level E
Inequalities
Level E Solving Inequalities

Examples : Choosing from -3, -2, -1, 0 , 1, 2, 3
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(a)   y<0                           -3, -2, -1

(b)   y>2                           3

(c)   y ≤ -2                       -3, -2

(d)   y>3                        No solution

25-Nov-11              Created by Mr.Lafferty Maths Dept
Level E
Inequalities
Level E Solving Inequalities
www.mathsrevision.com

Now try Exercise 6
Ch8 (page 99)

25-Nov-11         Created by Mr.Lafferty Maths Dept
Level E
Starter Questions

1. Simplify 10 y 2  2 y  8 y 2  3 y  3 y
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2. The ratio of apples to pears is 5 :14.
If there are 57 pieces of fruit in total.
How many pears are there.

3.     Find LCM of 6 and 10

25-Nov-11             Created by Mr.Lafferty Maths Dept
Level E
Inequalities
Level E Inequalities

Learning Intention                                     Success Criteria
www.mathsrevision.com

1. To solve simple inequalities                     1. Know rule
using the same rules as for                               „change side change sign‟.
equations
„change side change sign‟.
2. Solving simple inequalities

equations.

25-Nov-11                    Created by Mr.Lafferty Maths Dept
Level E
Inequalities
Level E Inequalities

The Good News
www.mathsrevision.com

Same rule as solving equations

The Balancing Method

25-Nov-11             Created by Mr.Lafferty Maths Dept
Level E
Inequalities
Level E Solving Inequalities

The Balancing Method
www.mathsrevision.com

Subtract
Add 9 to
10 from
each side
Examples                                                each side

x  9  16                                x  10  19
 x  16  9                             x  19  10
 x  25                                  x9
Check to see if correct !                Check to see if correct !
25-Nov-11           Created by Mr.Lafferty Maths Dept
Level E
Inequalities
Level E Solving Inequalities

The Balancing Method
www.mathsrevision.com

Divide by
Divide by                                 9 each
7 each                                     side
Examples           side
7 y  35                                      9m  36
35                                         36
 y                                         m
7                                          9
 y5                                       m4
Check to see if correct !                  Check to see if correct !
25-Nov-11           Created by Mr.Lafferty Maths Dept
Level E
Inequalities
Level E Inequalities
www.mathsrevision.com

Now try Exercise 7
Ch8 (page 101)

25-Nov-11         Created by Mr.Lafferty Maths Dept

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